9 How do we add and subtract fractions with like denominators?

Lesson
9 Adding and Subtracting Fractions
Problem Solving:
Making a Line Graph
Adding and Subtracting Fractions
How do we add and subtract fractions with like
denominators?
This fraction bar represents the fraction 3
4.
3
4
We identify the denominator by counting the total fractional parts.
There are 4 total parts. We identify the numerator by counting the
shaded parts. There are 3 shaded parts.
Adding Fractions With Like Denominators
We can use this idea of counting fractional parts to add fractions with
the same denominator. After all, we are just counting out more of the
same fractional part—thirds, fourths, fifths, tenths, and so on.
1 2
4+4
First, we look to see if both fractions have the same denominator.
Both fractions are fourths, so we are counting the same fractional
part–fourths.
1
4
+2
4
3
4
1 2 3
4+4=4
Remember, we can add these fractions because the denominators are
the same.
42 Unit 1 • Lesson 9
Lesson 9
When the denominators are the same, we know the fair shares are the
same. Let’s look at some examples using different pictures.
Example 1
Add the following fractions.
1 2
6+6
+
When we add or
subtract fractions, the
denominators have to
be the same.
=
The answer is 3
6.
3 4
8+8
+
=
The answer is 7
8.
Subtracting Fractions With Like Denominators
We also must have the same denominator when we subtract fractions.
The fair shares have to be the same. The need for fair shares is easier to
see with subtraction because we take away the fractional part.
4 1
5−5
4 1
55
3
5
4 1 3
5−5=5
The answer is 3
5.
Unit 1 • Lesson 9 43
Lesson 9
How do we add and subtract fractions with
unlike denominators?
Below is a fable about two peasants and a king. It helps us think about
the problem of adding and subtracting fractions. It also shows us why
the denominators have to be the same. If they are not the same, we
have to find equivalent fractions.
Hundreds of years ago, p eople didn’t always use m oney to buy the things they
needed. They would trade work f or food or a p lace to live. One day a k ing
was giving out bread to the peasants who had d one some w ork f or h im. Two
1
peasants came t o get their bread. Peasant George got 3 of a l oaf of bread,
1
and Peasant Dane got 2 of a l oaf of bread. George asked the king, “ Why did
I get less bread?” The king said, “ If you tell me e xactly h ow m uch less bread
you were given, I will give you the same a mount as your f riend.” This was a b ig
problem f or George, b ecause h e had to f igure out exactly what the difference
1
1
was between 2 and 3 .
Exactly how much
smaller is George’s
loaf than Dane’s?
This illustration shows
1
us the problem when we
2
1
try to find the difference between 1
2 and 3 using fraction bars. We can
see the difference, but we do not know exactly what it is. The difference
is not a fair share of either of the fraction bars.
1
What is the difference between 1
2 and 3 ?
To solve this problem, we need to change the denominators for
both fractions.
44 Unit 1 • Lesson 9
1
3
Lesson 9
1
We need to change 1
2 and 3 so that they have the same denominator.
We can do this using fraction bars. We use the fraction bar for sixths
because 6 is the least common multiple for both 2 and 3. We can make
1
equivalent fractions that meet at 1
2 and 3 .
The fraction bar for sixths lines up with the fraction bars for halves
1
and thirds. We can find an equivalent fraction for both 1
2 and 3
on the fraction bar for sixths.
3
1
2
Example 1 shows that we can change 1
2 into 6 and 3 into 6 .
Example 1
Find an equivalent fraction for both 12 and 13 .
3
Change 1
2 into 6 using fraction bars.
Using another fraction bar is one way
to find denominators that are the same.
halves
1
2
= 36
sixths
2
Change 1
3 into 6 using fraction bars.
thirds
1
3
= 26
sixths
2 1
1
Now we can rewrite the problem: 3
6 − 6 = 6 . George received 6 less bread than Dane.
Apply Skills
Reinforce Understanding
Turn to Interactive Text,
page 25.
Use the mBook Study Guide
to review lesson concepts.
Unit 1 • Lesson 9 45
Lesson 9
Problem Solving: Making a Line Graph
How do we make a line graph?
Bounce Height (in feet)
We have looked at what a line graph is. Now we will look at how to use
a line graph. A line graph is a good graph to use to record the data
from the following story.
1
16
2
12
3
10
A science class wants to measure the height of the bounce a ball
makes when it is dropped from 20 feet. Each time a ball bounces, the
height of the bounce gets smaller until the ball stops bouncing. The
students drop a golf ball onto a concrete floor and keep track of how
high the ball bounces for 10 bounces.
4
7
5
6
6
5
7
4
8
3
9
2
10
1
Now that we have a table of data, we can take this information and
make a line graph. Remember that a line graph compares two sets
of data.
Steps for Making a Line Graph
How High the Ball Could Bounce
Step 2
Label the vertical axis (the line on the left) “Height.”
Mark off the points on this axis from 0 to 20
because the height of the biggest bounce was close
to 20.
20
18
16
14
12
Height
Step 1
Draw the axes. Label the horizontal axis (the line at
the bottom) “Bounce.” Mark off the points on this
axis from 0 to 10 because there are 10 bounces
to record.
10
8
6
4
2
0
1
2
Step 3
We also need to write a title for the graph: “How
High the Ball Could Bounce.”
Step 4
Now we record the data. Start with the first bounce. Locate the height,
and plot a point. Continue with the second point, and so on.
46 Problem-Solving Activity
Reinforce Understanding
Turn to Interactive Text,
page 26.
Use the mBook Study Guide
to review lesson concepts.
Unit 1 • Lesson 9
3
4
5
6
Bounce
7
8
9
10
Lesson 9
Homework
Activity 1
Solve the problems involving fractions with like denominators.
1
3
+ 13
3
1
3.
4
5
− 25
4.
2
3
− 13
5.
3
9
+ 29
7
6
1.
2. 5 + 5
2
3
2
5
5
9
6. 8 − 8
4
5
1
3
1
8
Activity 2
Use the fraction bars to help you solve the problems with unlike denominators.
1.
1
2
+ 13
5
6
1
1
3
4
2
3
− 14
2. 2 + 4
3.
5
12
Activity 3 • Distributed Practice
Solve.
1.
1,200
 800
400
2.
6,701
+ 2,199
8,900
3.
895
9
8,055
4.
37
46
5. 9q841
1,702
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Unit 1 • Lesson 9 47